Lesson Angle in a semicircle

Source code of 'Angle in a semicircle'

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Theorem: An angle inscribed in a <A HREF=Semi_circle.wikipedia>Semi-circle</A> is a right angle.

{{{drawing( 260, 260, -1, 6, -1, 5.0, line( -.5, 2,4.5,2),line( .8, 3.9,2,2), line( .8, 3.9,-.5,2),line( .8, 3.9,4.5,2), locate( .75, 3.7,x),locate( 1.2, 3.7,y), locate( 3.6,2.4, y),locate( -.1,2.4,x),locate( -.75,2,B),locate( 2,2,C),locate( 2.1,2.4,z),locate( 1.5,2.4,w),locate( 4.6,2,D), locate( .75,4.3,A), red(circle( 2, 2,.1 )), circle( 2,2,2.5))}}}

In the above diagram, We have a circle with center 'C' and radius AC=BC=CD. Angle inscribed in 
semi-circle is angle BAD.

To proof this theorem,

Required construction is shown in the diagram. Draw the lines AB, AD and AC.
Now there are three triangles ABC, ACD and ABD.
Angles of the triangles are shown in the diagram.

Observe that triangle ABC and ACD are <A HREF=Isosceles_triangle.wikipedia>Isosceles triangle</A>. 
In triangle ABC
{{{BC=AC}}}
Angles in front of these lines are also same (Call it x)............(Property of isosceles triangle)

Similarly, 
In triangle ACD
{{{AC=DC}}}
Angles in front of these lines are also same (Call it y)............(Property of isosceles triangle)

Now again consider triangle ABC:
Sum of all three angles in the triangle is 180.............................(Property of triangle)
Hence {{{x+ y + w= 180}}}
{{{w = 180-(x+ y)}}}..........................................................(1)

Similarly, 
In triangle ACD:
{{{x+ y + z= 180}}}
{{{z = 180-(x+ y)}}}..........................................................(2)

After adding equation number (1) and (2),

{{{w + z= 360 - 2*(x+y)}}}..................................................(3)

Observe, BCD is a straight line. Hence, Angle(BCD) is a <A HREF=Straight_angle.wikipedia>straight angle</A>.

{{{w + z = 180}}}.............................................................................................(Property of Straight Angle)

Now, plug the valve of Angle(w+z) in equation number (3)

{{{180 = 360 - 2*(x+y)}}}
{{{2*(x+y)=180}}}
{{{x+y =90}}}

Hence, Angle(BAD) is <A HREF=Right_angles.wikipedia>right angle</A>.

Proof Ends!!!

For more information on Circle refer to <A HREF=Circle.wikipedia>wikipedia</A>.